An Analytic Solution for the Frontal Flow Period in 1D Counter-Current Spontaneous Imbibition into Fractured Porous Media Including Gravity and Wettability Effects

Including gravity and wettability effects, a full analytical solution for the frontal flow period for 1D counter-current spontaneous imbibition of a wetting phase into a porous medium saturated initially with non-wetting phase at initial wetting phase saturation is presented. The analytical solution applicable for liquid–liquid and liquid–gas systems is essentially valid for the cases when the gravity forces are relatively large and before the wetting phase front hits the no-flow boundary in the capillary-dominated regime. The new analytical solution free of any arbitrary parameters can also be utilized for predicting non-wetting phase recovery by spontaneous imbibition. In addition, a new dimensionless time equation for predicting dimensionless distances travelled by the wetting phase front versus dimensionless time is presented. Dimensionless distance travelled by the waterfront versus time was calculated varying the non-wetting phase viscosity between 1 and 100 mPas. The new dimensionless time expression was able to perfectly scale all these calculated dimensionless distance versus time responses into one single curve confirming the ability for the new scaling equation to properly account for variations in non-wetting phase viscosities. The dimensionless stabilization time, defined as the time at which the capillary forces are balanced by the gravity forces, was calculated to be approximately 0.6. The full analytical solution was finally used to derive a new transfer function with application to dual-porosity simulation.     

Pore-Scale Modelling of Rate Effects in Waterflooding

We develop a rate-dependent network model that accounts for viscous forces by solving for the wetting and non-wetting phase pressure and which allows wetting layer swelling near an advancing flood front. The model incorporates a new time-dependent algorithm by accounting for partial filling of elements. We use the model to study the effects of capillary number, mobility ratio and contact angle distribution on waterflood displacement patterns, saturation and velocity profiles. By using large networks, generated from a new stochastic network algorithm, we reproduce Buckley–Leverett profiles directly from pore-scale modelling thereby providing a bridge between pore-scale and macro-scale transport.     

Capillary Pressure at the Imbibition Front During Water–Oil Counter-Current Spontaneous Imbibition

Counter-current spontaneous imbibition (COUCSI) in porous media is driven by capillary forces. Capillary action results in a high capillary imbibition pressure at the imbibition front and a low capillary drainage pressure at the outlet face. It is the difference between these two pressures that draws in the wetting phase and pushes out the non-wetting phase. A technique for measuring the capillary pressure at an imbibition front under restricted flow conditions has been developed and applied to Berea sandstone with a range of permeabilities. In the experiments, brine was the wetting phase and refined oil was the non-wetting phase. One end face of a sandstone core was butted to a short section of a finer-pored rock. The composite core surface was then sealed, apart from the end face of the low-permeability segment. A connection to a pressure transducer was set in the opposite end face of the core. Initially, the main core segment was filled with oil. In most cases, the finer-pored segment was filled with brine. Imbibition was started by immersing the core in brine. The purpose of the finer-pored segment was to prevent the escape of non-wetting phase from the open face. For some tests there was an initial period of co-current spontaneous imbibition (COCSI) created by allowing production of non-wetting phase through an outlet tapping in the sealed end face. The outlet was then connected to the transducer and the imbibition changed to COUCSI. There followed an increase in the monitored end pressure to a maximum as fluid redistributed within the core. For the tests in which the fine-pored segments were pre-saturated with brine, even without an initial period of co-current imbibition, limited invasion of the main core segment by brine resulted in an asymptotic rise of the end pressure to a maximum as the imbibition front dispersed. To confirm that the dispersing front did not reach the dead end of the core, the distance of advance of the wetting liquid was detected by a series of electrodes. The maximum value of the end pressure provides an estimate of the capillary pressure at an imbibition front for COUCSI. The maximum capillary pressure generated by the invading fluids ranged from 6.6 kPa to 42 kPa for sandstone with permeabilities between 1.050 (μm)² and 0.06 (μm)².     

Condition for Break-up of Non-Wetting Fluids in Sinusoidally Constricted Capillary Channels

Analysis of capillary-pressure distribution in single channels with sinusoidal profile shows that surface tension-driven flow in such channels is controlled by the pressure extrema at their “crests” and “troughs”. Formulating the geometric condition for the pressure in the troughs to exceed that in the crests leads to a simple criterion for the spontaneous break-up of the non-wetting fluid in the necks of the constrictions. The criterion reduces to the condition for the Plateau-Rayleigh instability as a limiting case. Similar pressure analysis is applicable to the case of a non-wetting fluid invading an open pore body. Computational-fluid-dynamics experiments have verified the validity of the break-up predicted from the capillary-pressure argument. Although the geometric criterion for the break-up is valid for small capillary numbers, it provides a common framework in which the results of various published studies of a non-wetting phase choke-off in capillary constrictions for a wide range of capillary numbers can be explained and understood.     

Relative permeability and capillary pressure functions of porous media as related to the displacement growth pattern

Visualization experiments of the unsteady immiscible displacement of a fluid by another are performed on glass-etched pore networks of well-controlled morphology by varying the fluid system and flow conditions. The measured transient responses of the fluid saturation and pressure drop across the porous medium are introduced into numerical solvers of the macroscopic two-phase flow equations to estimate the non-wetting phase, k_rnw, and wetting phase, k_rw, relative permeability curves and capillary pressure, P_c, curve. The correlation of k_rnw, k_rw, and P_c with the displacement growth pattern is investigated. Except for the capillary number, wettability, and viscosity ratio, the immiscible displacement growth pattern in a porous medium may be governed by the shear-thinning rheology of the injected or displaced fluid, and the porous sample length as compared to the thickness of the frontal region. The imbibition k_rnw increases as the flow pattern changes from compact displacement to viscous fingering or from viscous to capillary fingering. The imbibition k_rw increases as the flow pattern changes from compact displacement or capillary fingering to viscous fingering. As the shear-thinning behaviour of the NWP strengthens and/or the contact angle decreases, then the flow pattern is gradually dominated by irregular interfacial configurations, and the imbibition k_rnw increases. The imbibition P_c is a decreasing function of the capillary number or increasing function of the injected phase viscosity in agreement with the linear thermodynamic theory.     

Capillary Heterogeneity Trapping and Crossflow in Layered Porous Media

We examine the effect of capillary and viscous forces on the displacement of one fluid by a second, immiscible fluid across and along parallel layers of contrasting porosity, and relative permeability, as well as previously explored contrasts in absolute permeability and capillary pressure. We consider displacements with wetting, intermediate-wetting and non-wetting injected phases. Flow is characterized using six independent dimensionless numbers and a dimensionless storage efficiency, which is numerically equivalent to the recovery efficiency. Results are directly applicable to geologic carbon storage and hydrocarbon production. We predict how the capillary–viscous force balance influences storage efficiency as a function of a small number of key dimensionless parameters, and provide a framework to support mechanistic interpretations of complex field or experimental data, and numerical model predictions, through the use of simple dimensionless models. When flow is directed across layers, we find that capillary heterogeneity traps the non-wetting phase, regardless of whether it is the injected or displaced phase. However, minimal trapping occurs when the injected phase is intermediate-wetting or when high-permeability layers contain a smaller moveable volume of fluid than low-permeability layers. A dimensionless capillary-to-viscous number defined using the layer thickness rather than the more commonly used system length is most relevant to predict capillary heterogeneity trapping. When flow is directed along layers, we show that, regardless of wettability, increasing capillary crossflow reduces the distance between the leading edges of the injected phase in each layer and increases storage efficiency. This may be counter-intuitive when the injected phase is non-wetting. Crossflow has a significant impact on storage efficiency only when high-permeability layers contain a smaller moveable volume of fluid than low-permeability layers. In that case, capillary heterogeneity traps the wetting phase, regardless of whether it is the injected or displaced phase.     

Characterization of the crossover from capillary invasion to viscous fingering to fracturing during drainage in a vertical 2D porous medium

We experimentally studied the displacement of a viscous wetting fluid (water) by an inviscid non-wetting fluid (air) injected at the bottom of a vertical Hele-Shaw cell filled with glass microbeads. In order to cover a wide parameter space, the permeability of the porous medium was varied by using different bead size ranges and diverse air flow rates were generated by means of a syringe pump. A LED light table was used to back illuminate the experimental cell, allowing a high speed camera to capture images of the drainage process at equal time intervals. The invasion occurred in intermittent bursts. Image processing of the bursts and fractal analysis showed successive transitions from capillary invasion to viscous fingering to fracturing during the same experiment, dependent on the medium permeability, the air injection flow rate, and the vertical position in the cell. The interplay between the capillary, viscous and gravity forces determines the nature of the invasion pattern and the transitions, from capillary invasion to viscous fingering with decreasing fluid pressure on one hand and from viscous fingering to fracturing with decreasing effective overburden pressure on the other hand.     

Effective Rheology of Two-Phase Flow in Three-Dimensional Porous Media: Experiment and Simulation

We present an experimental and numerical study of immiscible two-phase flow of Newtonian fluids in three-dimensional (3D) porous media to find the relationship between the volumetric flow rate (Q) and the total pressure difference (\(\Delta P\)) in the steady state. We show that in the regime where capillary forces compete with the viscous forces, the distribution of capillary barriers at the interfaces effectively creates a yield threshold (\(P_t\)), making the fluids reminiscent of a Bingham viscoplastic fluid in the porous medium. In this regime, Q depends quadratically on an excess pressure drop (\(\Delta P-P_t\)). While increasing the flow rate, there is a transition, beyond which the overall flow is Newtonian and the relationship is linear. In our experiments, we build a model porous medium using a column of glass beads transporting two fluids, deionized water and air. For the numerical study, reconstructed 3D pore networks from real core samples are considered and the transport of wetting and non-wetting fluids through the network is modeled by tracking the fluid interfaces with time. We find agreement between our numerical and experimental results. Our results match with the mean-field results reported earlier.     

Review of Steady-State Two-Phase Flow in Porous Media: Independent Variables,Universal Energy Efficiency Map,Critical Flow Conditions,Effective Characterization of Flow and Pore Network

In many applications of two-phase flow in porous media, a wetting phase is used to displace through a network of pore conduits as much as possible of a non-wetting phase, residing in situ. The energy efficiency of this physical process may be assessed by the ratio of the flow rate of the non-wetting phase over the total mechanical power externally provided and irreversibly dissipated within the process. Fractional flow analysis, extensive simulations implementing the DeProF mechanistic model, as well as a recent retrospective examination of laboratory studies have revealed universal systematic trends of the energy efficiency in terms of the actual independent variables of the process, namely the capillary number, Ca, and the flow rate ratio, r. These trends can be cast into an energy efficiency map over the (Ca, r) domain of independent variables. The map is universal for all types of non-wetting/wetting phase porous medium systems. It demarcates the efficiency of steady-state two-phase flow processes in terms of pertinent system parameters. The map can be used as a tool for designing more efficient processes, as well as for the normative characterization of two-phase flows, as to the predominance of capillary or viscous effects. This concept is based on the existence of a unique locus of critical flow conditions, for which the energy efficiency takes locally maximum values. The locus shape depends on the physicochemical characteristics of the non-wetting phase/wetting phase/porous medium system, and it shows a significant mutation as the externally imposed flow conditions change the type of flow, from capillary- to viscosity-dominated. The locus can be approached by an S-type functional form in terms of the capillary number and the system properties (viscosity ratio, wettability, pore network geometry, etc.), suggesting that formative criteria can be derived for flow characterization in any system. A new, extended definition of the capillary number is also proposed that effectively takes into account the critical properties of all the system constituents. When loci of critical flow conditions pertaining to processes with different viscosity ratio in the same pore network, are expressed in terms of this true-to-mechanism capillary number, they collapse into a unique locus. In this context, a new methodology for the effective characterization of pore networks is proposed.     

Influence of Numerical Cementation on Multiphase Displacement in Rough Fractures

We present an application of 3D X-ray computed microtomography for studying the influence of numerical cementation on flow in a cement-lined rough-walled fracture. The imaged fracture geometry serves as input for flow modeling using a combination of the level set and the lattice Boltzmann methods to characterize the capillary-dominated fluid displacement properties and the relative permeability of the naturally cemented fracture. We further numerically add cement to the naturally cement-lined fracture to quantify the effect of increasing cement thickness and diminishing aperture on flow properties. Pore space geometric tortuosity and capillary pressure as a function of water saturation both increase with the numerically increased fracture cement thickness. The creation of unevenly distributed apertures and cement contact points during numerical cement growth causes the wetting and non-wetting fluids to impede each other, with no consistent trends in relative permeability with increasing saturation. Tortuosity of wetting and non-wetting fluid phases exhibits none to poor correlation with relative permeability and thus cannot be used to predict it, contrary to previous findings in smoother fractures.     

Microscale Visual Study of End Effects at Permeability Discontinuities

The physical effect of multiphase fluid distribution and flow at permeability boundaries has not been fully investigated, particularly at the pore scale (1–100 μm), although such behaviour can significantly affect the overall scaled-up reservoir trapping capacity and production performance. In this article, microscale physical models have been used to qualitatively study the pore scale flow events at permeability boundaries, both high to low and vice versa, to gain a better understanding of the role of these boundaries and water saturation on multiphase displacement behaviour at the pore scale. We have used etched glass models of stripes of large and small (a factor of two) pores with circular matrix. Capillary pressure, which is the controlling parameter is itself dependant on pore size and its spatial distribution, the magnitude of the interfacial tensions and the wettability between the fluids and the solid surface of the models. Sometimes, the only way the non-wetting fluid can penetrate the boundary is through a fortuitous leakage, whereby the presence of an initial saturation reduces the controlling capillary pressure. Examples are demonstrated including mechanisms of end-effects and how capillary boundary resistance (due to capillary forces) can be broken down and fluid movement across the boundary can develop. These micromodel experiments show vividly that connate water can assist in these processes, particularly oil trapping and leakage of water across a permeability boundary.     

Analysis of Seepage for Power-Law Fluids in the Fractal-Like Tree Network

This work studies the flow characteristics of power-law fluids in the fractal-like tree network. A fractal model is developed for the permeability of power-law fluid flow in fractal-like tree network based on straight capillary model, generalized Darcy’s law and constitutive equation for power-law fluids. Analytical expression for permeability of power-law fluids in the network is presented and found to be a function of network microstructural parameters such as the branching diameter ratio, the branching length ratio, the total number of branching levels, the bifurcation angle, the branching number, the diameter of the zeroth branching level and the power exponent of power-law fluids. Both the phase permeabilities and the relative permeabilities are also derived and found to be a function of power exponent for the wetting phase and non-wetting phase, the saturation and other microstructural parameters and independent of the bifurcation angle.     

Elliptic Regions and Stable Solutions for Three-Phase flow in Porous Media

In the limit of zero capillary pressure, solutions to the equations governing three-phase flow, obtained using common empirical relative permeability models, exhibit complex wavespeeds for certain saturation values (elliptic regions) that result in unstable and non-unique solutions. We analyze a simple but physically realizable pore-scale model: a bundle of cylindrical capillary tubes, to investigate whether the presence of these elliptic regions is an artifact of using unphysical relative permeabilities. Without gravity, the model does not yield elliptic regions unless the most non-wetting phase is the most viscous and the most wetting phase is the least viscous. With gravity, the model yields elliptic regions for any combination of viscosities, and these regions occupy a significant fraction of the saturation space. We then present converged, stable numerical solutions for one-dimensional flow, which include capillary pressure. These demonstrate that, even when capillary forces are small relative to viscous forces, they have a significant effect on solutions which cross or enter the elliptic region. We conclude that elliptic regions can occur for a physically realizable model of a porous medium, and that capillary pressure should be included explicitly in three-phase numerical simulators to obtain stable, physically meaningful solutions which reproduce the correct sequence of saturation changes.     

Seismic Reflection from an Interface Between an Elastic Solid and a Fractured Porous Medium with Partial Saturation

The theory of Tuncay and Corapcioglu (Transp Porous Media 23:237–258, 1996a) has been employed to investigate the possibility of plane wave propagation in a fractured porous medium containing two immiscible fluids. Solid phase of the porous medium is assumed to be linearly elastic, isotropic and the fractures are assumed to be distributed isotropically throughout the medium. It has been shown that there can exist four compressional waves and one rotational wave. The phase speeds of these waves are found to be affected by the presence of fractures, in general. Of the four compressional waves, one arises due to the presence of fractures in the medium and the remaining three are those encountered by Tuncay and Corapcioglu (J Appl Mech 64:313–319, 1997). Reflection and transmission phenomena at a plane interface between a uniform elastic half-space and a fractured porous half-space containing two immiscible fluids, are analyzed due to incidence of plane longitudinal/transverse wave from uniform elastic half-space. Variation of modulus of amplitude and energy ratios with the angle of incidence are computed numerically by taking the elastic half-space as granite and the fractured porous half-space as sandstone material containing non-viscous wetting and non-wetting fluid phases. The results obtained in case of porous half-space with fractures, are compared graphically with those in case of porous half-space without fractures. It is found that the presence of fractures in the porous half-space do affect the reflection/transmission of waves, which is responsible for raising the reflection and lowering the transmission coefficients.     

Insights into the Relationships Among Capillary Pressure,Saturation, Interfacial Area and Relative Permeability Using Pore-Network Modeling

To gain insight in relationships among capillary pressure, interfacial area, saturation, and relative permeability in two-phase flow in porous media, we have developed two types of pore-network models. The first one, called tube model, has only one element type, namely pore throats. The second one is a sphere-and-tube model with both pore bodies and pore throats. We have shown that the two models produce distinctly different curves for capillary pressure and relative permeability. In particular, we find that the tube model cannot reproduce hysteresis. We have investigated some basic issues such as effect of network size, network dimension, and different trapping assumptions in the two networks. We have also obtained curves of fluid–fluid interfacial area versus saturation. We show that the trend of relationship between interfacial area and saturation is largely influenced by trapping assumptions. Through simulating primary and scanning drainage and imbibition cycles, we have generated two surfaces fitted to capillary pressure, saturation, and interfacial area (P _c –S _w –a _nw ) points as well as to relative permeability, saturation, and interfacial area (k _r –S _w –a _nw ) points. The two fitted three-dimensional surfaces show very good correlation with the data points. We have fitted two different surfaces to P _c –S _w –a _nw points for drainage and imbibition separately. The two surfaces do not completely coincide. But, their mean absolute difference decreases with increasing overlap in the statistical distributions of pore bodies and pore throats. We have shown that interfacial area can be considered as an essential variable for diminishing or eliminating the hysteresis observed in capillary pressure–saturation (P _c –S _w ) and the relative permeability–saturation (k _r –S _w ) curves.     

Averaged model with capillary nonequilibrium effects for two-phase flow through a highly heterogeneous porous medium

Two-phase flow through a medium with two porosities in which the absolute permeabilities and the capillary pressure functions of the components differ by an order of magnitude is investigated. A classification and diagram of the elementary flows are proposed at the single cell level. An averaged model is developed for a single class of systems in which source-type capillary-dispersion flow predominates in the blocks. This model contains a nonlinear kinetic relation between the average values of the capillary pressure functions. An expansion of the effective phase permeability tensor allowing it to be calculated efficiently is proposed. The capillary relaxation time is explicitly determined. Examples of calculations of the averaged phase permeability tensor and the capillary relaxation time are given. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 93–103, May–June, 1998. The work was carried out with the support from the Russian Foundation for Basic Research (project No. 95-01-01179a).     

Volume Conservation of the Intermediate Phase in Three-Phase Pore-Network Models

Quasi-static rule-based network models used to calculate capillary dominated multi-phase transport properties in porous media employ equilibrium fluid saturation distributions which assume that pores are fully filled with a single bulk fluid with other fluids present only as wetting and/or spreading films. We show that for drainage dominated three-phase displacements in which a non-wetting fluid (gas) displaces a trapped intermediate fluid (residual oil) in the presence of a mobile wetting fluid (water) this assumption distorts the dynamics of three-phase displacements and results in significant volume errors for the intermediate fluid and erroneous calculations of intermediate fluid residual saturations, relative permeabilities and recoveries. The volume errors are associated with the double drainage mechanism which is responsible for the mobilization of waterflood residual oil. A simple modification of the double drainage mechanism is proposed which allows the presence of a relatively small number of partially filled pores and removes the oil volume errors.     

On the Structure and Flow Processes in the Capillary Fringe of Phreatic Aquifers

The water distribution in the capillary fringe (CF) reflects the interaction of a strongly wetting fluid in a heterogeneous porous medium. Field profiles of gravimetric water content of the CF for a 30m deep, sandy, phreatic aquifer in Israel are critically analyzed in the context of the possible wetting and drainage processes in these sediments. A highly plausible explanation of the profiles is based on the spatial configuration of the CF surface determined from a model of the movement of water within the porous medium. The structural types of CF that can arise from a number of competing pore-scale displacement mechanisms, in the presence of gravity, are characterized by the model. We differentiate between two generic types of CF structures: a tenuous invasion-percolation type and a compact type. Flow, in response to a horizontal pressure gradient, associated with each structure is analyzed. Our interpretation of the field data supports the compact structure with a spatial variation in the height of the CF surface, above the water table, on the order of 1m. In this compact structure horizontal flow is characterized by stagnant regions in the CF above a critical height h _c and flow only for regions below h _c . The field water content (at h _c ) may be used to predict the onset of lateral water flow in the CF.